scholarly journals The Unique Games Conjecture, Integrality Gap for Cut Problems and Embeddability of Negative-Type Metrics into ℓ 1

2015 ◽  
Vol 62 (1) ◽  
pp. 1-39 ◽  
Author(s):  
Subhash A. Khot ◽  
Nisheeth K. Vishnoi
Keyword(s):  
Integrality Gap ◽  
Negative Type ◽  
Unique Games Conjecture ◽  
Unique Games ◽  
Cut Problems

scholarly journals Three candidate plurality is stablest for small correlations

2021 ◽  
Vol 9 ◽  
Author(s):  
Steven Heilman ◽  
Alex Tarter
Keyword(s):  
Structure Theorem ◽  
Semidefinite Program ◽  
Dimensional Euclidean Space ◽  
List Type ◽  
Noise Stability ◽  
Equal Chance ◽  
Unique Games Conjecture ◽  
Fixed Constant ◽  
Correlation Parameters ◽  
Unique Games

Abstract Using the calculus of variations, we prove the following structure theorem for noise-stable partitions: a partition of n-dimensional Euclidean space into m disjoint sets of fixed Gaussian volumes that maximise their noise stability must be $(m-1)$ -dimensional, if $m-1\leq n$ . In particular, the maximum noise stability of a partition of m sets in $\mathbb {R}^{n}$ of fixed Gaussian volumes is constant for all n satisfying $n\geq m-1$ . From this result, we obtain: (i) A proof of the plurality is stablest conjecture for three candidate elections, for all correlation parameters $\rho $ satisfying $0<\rho <\rho _{0}$ , where $\rho _{0}>0$ is a fixed constant (that does not depend on the dimension n), when each candidate has an equal chance of winning. (ii) A variational proof of Borell’s inequality (corresponding to the case $m=2$ ). The structure theorem answers a question of De–Mossel–Neeman and of Ghazi–Kamath–Raghavendra. Item (i) is the first proof of any case of the plurality is stablest conjecture of Khot-Kindler-Mossel-O’Donnell for fixed $\rho $ , with the case $\rho \to L1^{-}$ being solved recently. Item (i) is also the first evidence for the optimality of the Frieze–Jerrum semidefinite program for solving MAX-3-CUT, assuming the unique games conjecture. Without the assumption that each candidate has an equal chance of winning in (i), the plurality is stablest conjecture is known to be false.

scholarly journals 2-Approximating Feedback Vertex Set in Tournaments

2021 ◽  
Vol 17 (2) ◽  
pp. 1-14
Author(s):  
Daniel Lokshtanov ◽  
Pranabendu Misra ◽  
Joydeep Mukherjee ◽  
Fahad Panolan ◽  
Geevarghese Philip ◽  
...  
Keyword(s):  
Approximation Algorithm ◽  
Weight Function ◽  
Directed Graph ◽  
Polynomial Time ◽  
Optimal Solutions ◽  
Feedback Vertex Set ◽  
Vertex Set ◽  
Feedback Vertex Set Problem ◽  
Unique Games Conjecture ◽  
Unique Games

A tournament is a directed graph T such that every pair of vertices is connected by an arc. A feedback vertex set is a set S of vertices in T such that T − S is acyclic. We consider the Feedback Vertex Set problem in tournaments. Here, the input is a tournament T and a weight function w : V ( T ) → N, and the task is to find a feedback vertex set S in T minimizing w ( S ) = ∑ v∈S w ( v ). Rounding optimal solutions to the natural LP-relaxation of this problem yields a simple 3-approximation algorithm. This has been improved to 2.5 by Cai et al. [SICOMP 2000], and subsequently to 7/3 by Mnich et al. [ESA 2016]. In this article, we give the first polynomial time factor 2-approximation algorithm for this problem. Assuming the Unique Games Conjecture, this is the best possible approximation ratio achievable in polynomial time.

scholarly journals The Quest for Strong Inapproximability Results with Perfect Completeness

2021 ◽  
Vol 17 (3) ◽  
pp. 1-35
Author(s):  
Joshua Brakensiek ◽  
Venkatesan Guruswami
Keyword(s):  
Constraint Satisfaction ◽  
Absolute Constant ◽  
Independent Set ◽  
Constant Factor ◽  
Constraint Satisfaction Problems ◽  
Uniform Hypergraph ◽  
Semidefinite Programming Relaxation ◽  
Worst Case ◽  
Unique Games Conjecture ◽  
Unique Games

The Unique Games Conjecture has pinned down the approximability of all constraint satisfaction problems (CSPs), showing that a natural semidefinite programming relaxation offers the optimal worst-case approximation ratio for any CSP. This elegant picture, however, does not apply for CSP instances that are perfectly satisfiable, due to the imperfect completeness inherent in the Unique Games Conjecture. This work is motivated by the pursuit of a better understanding of the approximability of perfectly satisfiable instances of CSPs. We prove that an “almost Unique” version of Label Cover can be approximated within a constant factor on satisfiable instances. Our main conceptual contribution is the formulation of a (hypergraph) version of Label Cover that we call V Label Cover . Assuming a conjecture concerning the inapproximability of V Label Cover on perfectly satisfiable instances, we prove the following implications: • There is an absolute constant c 0 such that for k ≥ 3, given a satisfiable instance of Boolean k -CSP, it is hard to find an assignment satisfying more than c 0 k 2 /2 k fraction of the constraints. • Given a k -uniform hypergraph, k ≥ 2, for all ε > 0, it is hard to tell if it is q -strongly colorable or has no independent set with an ε fraction of vertices, where q =⌈ k +√ k -1/2⌉. • Given a k -uniform hypergraph, k ≥ 3, for all ε > 0, it is hard to tell if it is ( k -1)-rainbow colorable or has no independent set with an ε fraction of vertices.

scholarly journals An approximability-related parameter on graphs―-properties and applications

2015 ◽  
Vol Vol. 17 no. 1 (Graph Theory) ◽  
Author(s):  
Robert Engström ◽  
Tommy Färnqvist ◽  
Peter Jonsson ◽  
Johan Thapper
Keyword(s):  
Graph Theory ◽  
Constraint Satisfaction ◽  
Constraint Satisfaction Problem ◽  
Complete Graphs ◽  
Arbitrary Graph ◽  
Symmetric Relation ◽  
Binary Parameter ◽  
International Audience ◽  
Unique Games Conjecture ◽  
Unique Games

Graph Theory International audience We introduce a binary parameter on optimisation problems called separation. The parameter is used to relate the approximation ratios of different optimisation problems; in other words, we can convert approximability (and non-approximability) result for one problem into (non)-approximability results for other problems. Our main application is the problem (weighted) maximum H-colourable subgraph (Max H-Col), which is a restriction of the general maximum constraint satisfaction problem (Max CSP) to a single, binary, and symmetric relation. Using known approximation ratios for Max k-cut, we obtain general asymptotic approximability results for Max H-Col for an arbitrary graph H. For several classes of graphs, we provide near-optimal results under the unique games conjecture. We also investigate separation as a graph parameter. In this vein, we study its properties on circular complete graphs. Furthermore, we establish a close connection to work by Šámal on cubical colourings of graphs. This connection shows that our parameter is closely related to a special type of chromatic number. We believe that this insight may turn out to be crucial for understanding the behaviour of the parameter, and in the longer term, for understanding the approximability of optimisation problems such as Max H-Col.

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